The Collatz Network

Abstract

Considering all possible paths that a natural number can take following the rules of the algorithm proposed in the Collatz conjecture we construct a graph that can be interpreted as an infinite network that contemplates all possible paths within the conjecture. This allows us to understand why the minimal element of the Collatz orbit $x$ is equal to 1. Subsequently we define the extended Collatz conjecture equal to $o x+1$ when $x$ is odd and $x/2$ when $x$ is even with $o$ an odd number greater than $3$ and we show that there are infinite orbits in the extended Collatz conjecture that diverges. Finally, we find interesting theorems relating the Fibonacci sequence to prime numbers.